This volume of the Encyclopaedia contains two contributions．In the first Yu.V.Egorov gives an accomnt of microlocal analysis as a tool for investigating partial differemial equations．This 113ethod has become increasingly important in the theory．of Hamiltonian systems in recent years．
The second survey written by V.Ya．1vrii treats linear hyperbolic equations and systems．The author states necessary and sufficiient conditions for C∞-and L2-well-posedness and he studies the analogous pmhlem in the

Chapter 1.Microlocal Properties of Distributions
2.Wave Front of Distribution.Its Functorial Properties
2.1.Definition ofthe Wave Front
2.2.Localization ofWave Front
2.3.Wave Front and Singularities of One—Dimensional
2.4.Wave Fronts of Pushforwards and Pullbacks of a
3.Wave Front and Operations on Distributions
3.1 The Trace of a Distribution.Product of Distnritbiaul Eiuation
3.2.The Wave Front of the Solution of a Differential Eqution
3.3.Wave Fronts and Integral Operators
Chapter 2.Pseudodifferential Operators
1.Algebra ofPseudodifferential Operators
1.1.Singular Integral Operators
1.2.The Symbol
1.3.Boundedness of Pseudodifferential Operators
1.4.Composition of Pseudodifferential Operators
1.5.The Formally Adjoint Operator
1.6.Pseudolocality.Microlocality
1.7.Elliptic Operators
1.8.Garding’S Inequality
1.9.Extension 0f the Class of Pseudodifferential Operators
2.Invariance of the Principal SymboJ Under Canonical Transformations
2.1.Invariance Under the Change ofVariables.
2.2 The Subprincipal Symbol
2.3.Canonical Transformations
2.4.An Inverse Theorem
3.Canonical Forms ofthe Symbol
3.1.Simple Characteristic Points
3.2.Double Characteristics
3.3.The Complex-alued Symbol
3.4.The Canonical Form of the Symbol in a Neighbourhood of the Boundary.
4.Various Classes of Pseudodifferential Operators
4.1.The Lm/pδClasses
4.2.The Lm/φ,φ Classes
4.3。The Weyl Operators
5.Complex Powers ofElliptic Operators
5.1.The Definition ofComplex Powers.
5.2.Thc Construction of the Symbol for the Operator Az
5.3.The Construction of the Kernel of the Operator Az
5.4.The ξ-Function ofan Elliptic Operator
5.5.The Asymptotics of the Spectral Function and Eigenvalues
5.6.Complex Powers of an Elliptic Operator with Boundary Conditions
6.Pseudodifferential Operators in IRn and Quantization
6.1.The Analogy Between the Microlocal Analysis and the Quantization
6.2.Pseudodifierential 0perators in Rn
Chapter 3.Fourier Integral Operators
1.The Parametrix of the Cauchy Problem for Hyperbolic Equations
1.1.The Cauchy Problem for the Wave Equation
1.2.The Cauchy Problem for the Hyperbolic Equation of an Arbitrary 0rder.
1.3.The Method of Stationary Phase
2.The Maslov Canonical Operator
2.1.The MaslOV Index
2.2.Pre.canonieal Operator
2.3.The Canonical Operator
2.4.Some Applications.
3.Fourier Integral Operators
3.1.The Oscillatory Integrals
3.2.The Local Definition of the Fourier Integral Operator
……
Chapter 4 The Propagation of Singularities
Chapter 5 Solvbility of (Pseudo)Differential Equations
Chapter 6 Smoothness of Solutions of Differential Equations
Chapter 7 Transformation of Boundary-Value Problems
Chapter 8 Hyperfuctions
References